In mathematics (especially algebraic topology and abstract algebra), homology (in part from Greek ὁμός ''homos'' "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group. See singular homology for a concrete version for topological spaces, or group cohomology for a concrete version for groups.
For a topological space, the homology groups are generally much easier to compute than the homotopy groups, and consequently one usually will have an easier time working with homology to aid in the classification of spaces.
The original motivation for defining homology groups is the observation that shapes are distinguished by their ''holes''. But because a hole is "not there", it is not immediately obvious how to define a hole, or how to distinguish between different kinds of holes. Homology is a rigorous mathematical method for defining and categorizing holes in a shape. It approaches the problem through the idea of ''cycles'' - closed loops or low-dimensional manifolds - that can be drawn on a given n dimensional manifold but not transformed smoothly into each other, for example because they pass through different holes. As it turns out, subtle kinds of holes exist that homology cannot "see" — in which case homotopy groups may be what is needed.
== Informal examples ==
Informally, the homology of a topological space ''X'' is a set of topological invariants of ''X'' represented by its ''homology groups''
where the homology group describes the ''k''-dimensional holes in ''X''. A 0-dimensional hole is simply a gap between two components, consequently describes the path-connected components of ''X''.
A one-dimensional sphere is a circle. It has a single connected component and a one-dimensional hole, but no higher-dimensional holes. The corresponding homology groups are given as
where is the group of integers and is the trivial group. The group represents a finitely-generated abelian group, with a single generator representing the one-dimensional hole contained in a circle.
A two-dimensional sphere has a single connected component, no one-dimensional holes, a two-dimensional hole, and no higher-dimensional holes. The corresponding homology groups are〔
In general for an ''n''-dimensional sphere ''Sn'', the homology groups are
A one-dimensional ball ''B''1 is a solid disc. It has a single path-connected component, but in contrast to the circle, has no one-dimensional or higher-dimensional holes. The corresponding homology groups are all trivial except for . In general, for an ''n''-dimensional ball ''Bn'',〔
The torus is defined as a Cartesian product of two circles . The torus has a single path-connected component, two independent one-dimensional holes (indicated by circles in red and blue) and one two-dimensional hole as the interior of the torus. The corresponding homology groups are
The two independent 1D holes form independent generators in a finitely-generated abelian group, expressed as the Cartesian product group .
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